The conjugate uniformization via 1-motives

We use the p-divisible group attached to a 1-motive to generalize the conjugate p-adic uniformization of Iovita–Morrow–Zaharescu to arbitrary p-adic formal semi-abelian schemes or p-divisible groups over the ring of integers in a p-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties.